Cyclic Hypergraph Product Code
Arda Aydin, Nicolas Delfosse, Edwin Tham
TL;DR
This work presents a cyclic-symmetry approach to quantum LDPC hypergraph product codes by constructing Cyclic HGP (CxC) codes from cyclic seed matrices ${\cal A}(x)$ and ${\cal B}(y)$. The authors define two high-performance families, C2 (symmetric product) and CxR (product with a repetition code), and perform an exhaustive search over seeds with $n_c\le 40$ and $2\le w\le 5$, yielding block length $n=2ab$ and logical qubits $k=2(a-r_a)(b-r_b)$ with distance $d_{\textsf{CxC}}=\min\{d_{\cal A},d_{\cal B}\}$. Circuit-level simulations show C2 codes achieving up to $\approx 2\times 10^{-8}$ logical error rate per logical qubit for instances like $[[882,50,10]]$C2, outperforming ML-optimized HGPs (e.g., $[[625,25,8]]$) by orders of magnitude, while some BB-code comparisons are regime-dependent; CxR codes remain competitive with lower-weight checks. The work also introduces a two-row cyclic QCCD-like layout enabling constant-depth syndrome extraction for LDPC codes, with explicit depth bounds $$(2w({\cal A})+2w({\cal B})+2)d$$ per $d$ rounds (and a non-modular variant with depth $(w({\cal A})+w({\cal B})+2)d+(w({\cal A})+1)$), highlighting practical, hardware-friendly implementations for trapped-ion and related platforms. Overall, the simple cyclic construction yields high-performance quantum LDPC codes with favorable circuit properties, suggesting a scalable path toward robust fault-tolerant quantum memory.
Abstract
Hypergraph product (HGP) codes are one of the most popular family of quantum low-density parity-check (LDPC) codes. Circuit-level simulations show that they can achieve the same logical error rate as surface codes with a reduced qubit overhead. They have been extensively optimized by importing classical techniques such as the progressive edge growth, or through random search, simulated annealing or reinforcement learning techniques. In this work, instead of machine learning (ML) algorithms that improve the code performance through local transformations, we impose additional global symmetries, that are hard to discover through ML, and we perform an exhaustive search. Precisely, we focus on the hypergraph product of two cyclic codes, which we call CxC codes and we study C2 codes which are the product a cyclic code with itself and CxR codes which are the product of a cyclic codes with a repetition code. We discover C2 codes and CxR codes that significantly outperform previously optimized HGP codes, achieving better parameters and a logical error rate per logical qubit that is up to three orders of magnitude better. Moreover, some C2 codes achieve simultaneously a lower logical error rate and a smaller qubit overhead than state-of-the-art LDPC codes such as the bivariate bicycle codes, at the price of a larger block length. Finally, leveraging the cyclic symmetry imposed on the codes, we design an efficient planar layout for the QCCD architecture, allowing for a trapped ion implementation of the syndrome extraction circuit in constant depth.